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Mushroom Math
The Mushroom Math is a complicated type of math taught in schools of the Mushroom Kingdom, Yoshi's Island, and rarely Island Dolphino. Examples Cooking Math One-up Mushroom + Star = Invincishroom Cooking Math is a type of Mushroom Kingdom math. This math is helpful for when you're an adult and you cook, because the math problems are recipes for food. The Cooking Math is really hard to master, and is not taught in Fifth Grade but it is taught in Private Schools. There is also the "Reverse Cooking Math", where it is like subtracting. Invincishroom - Star = One Up Mushroom Bigger Than Mega Mushroom > Mini Mushroom < Mushroom and One Up Mushroom = Mushroom Easy kind of math, this kind of math says what kind of Mushroom is bigger than what kind of Mushroom. This math also tells what is bigger-a Fire Flower or Mushroom, or other kind of items like Stars. Advanced Topics There are also advanced topics in Mushroom Math that are used by highly educated members of the Mushroom Kingdom. Group theory is an example. Let \mathfrak{G} be a set endowed with a binary operation \cdot : \mathfrak{G} \times \mathfrak{G} \longrightarrow \mathfrak{G} . Then \mathfrak{G} is a group if for all \mathfrak{g}, \mathfrak{h}, \mathfrak{k} \in \mathfrak{G} we have (\mathfrak{g} \cdot \mathfrak{h}) \cdot \mathfrak{k} = \mathfrak{g} \cdot (\mathfrak{h} \cdot \mathfrak{k}), \forall \mathfrak{x} \in \mathfrak{G}, \exists \mathfrak{x}^{-1} such that \mathfrak{x} \cdot \mathfrak{x}^{-1} = \mathfrak{x}^{-1} \cdot \mathfrak{x} = \mathfrak{e} where \mathfrak{e} \in \mathfrak{G} and \mathfrak{g} \cdot \mathfrak{e} = \mathfrak{e} \cdot \mathfrak{g} = \mathfrak{g} for all g \in \mathfrak{G}. From here we shall omit the operation \cdot, and when we write ab, we mean a \cdot b. Now some interesting properties of groups are that inverses and the identity \mathfrak{e} are unique. To see this let \mathfrak{e}_1 and \mathfrak{e}_2 be two identity elements. From the properites of identity elements we can see that \mathfrak{e}_1\mathfrak{e}_2 = \mathfrak{e}_1 = \mathfrak{e}_2, so they are indeed the same. Similarly let x \in \mathfrak{G}, and let \mathfrak{x}_1^{-1} and \mathfrak{x}_2^{-1} be two inverses of \mathfrak{x}. The inverse elements have the property that \mathfrak{x}\mathfrak{x}_1^{-1} = \mathfrak{e}, and we may multiply by \mathfrak{x}_2^{-1} to find that \mathfrak{x}_2^{-1}\mathfrak{x}\mathfrak{x}_1^{-1} = \mathfrak{e}\mathfrak{x}_1^{-1} = \mathfrak{x}_1^{-1} = \mathfrak{e}\mathfrak{x}_2^{-1} = \mathfrak{x}_2^{-1}, and the two inverses are equivalent. Another property of inverses is that (\mathfrak{x}^{-1})^{-1} = \mathfrak{x} for all \mathfrak{x} \in \mathfrak{G} because \mathfrak{x}\mathfrak{x}^{-1} = \mathfrak{x}^{-1}\mathfrak{x} = \mathfrak{e}. An example of a use of group theory is in mushroom cooking with the set of ingredients \mathfrak{H} = \{��,��,��,�� \}. We may express the relationships between group elements using a Cayley table. Notice that for all \mathfrak{h} \in \mathfrak{H}, we have ��\mathfrak{h} = \mathfrak{h}�� = \mathfrak{h}. This means that �� is the identity element which we referred to earlier as \mathfrak{e}. We can also notice that for all \mathfrak{g} \in \mathfrak{H} the relation \mathfrak{g}\mathfrak{g} = �� holds, so every element is its own inverse. Additionally, this group has the property that for all \mathfrak{x}, \mathfrak{y} \in \mathfrak{H}, \mathfrak{x} and \mathfrak{y} commute (ie \mathfrak{x}\mathfrak{y} = \mathfrak{y}\mathfrak{x}. This property does not hold in general for all groups, but when this property does hold for a group, the group is said to be abelian. Note that the group H is isomorphic to the Klein four-group. Two groups \mathfrak{A} and \mathfrak{B} are isomorphic if there exists a bijective function \phi between the two groups where \phi is a homomorphism. A function f : A \longrightarrow B is bijective when for all a_1, a_2 \in A we have f(a_1) = f(a_2) if and only if a_1 = a_2, and if for all b \in B there exists an a \in A such that f(a) = b. A function g : A \longrightarrow B is a homomorphism if for all a_1, a_2 \in A we have g(a_1)g(a_2) = g(a_1a_2). If two groups \mathfrak{G} and \mathfrak{H} are isomorphic, we write \mathfrak{G} \cong \mathfrak{H}. An isomorphism between two groups mean that the two groups are algebraically identical. Trivia *The game Mario's Math released in 1820 teaches this math. Also Super Spamming Paper Mario has this math in the game. *There's also a game with Powerup Algebra in it called Goomba Algebra. Category:Terms